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Topology/Geometry Seminar
Thursday, February 10, 2005
5:00-6:00pm,
122 Pond Lab (UP)
Christina Sormani
Lehman College, CUNY
Gromov-Hausdorff Stability of Schur's Lemma
Abstract: A Riemannian manifold is locally isotropic if the length of the third side of a small triangle is determined by the lengths of the first two sides, their common vertex, and the angle between the sides. This implies that the sectional curvature of the manifold depends only on the base point. By Schur's Lemma, it then follows that the sectional curvature is constant and the manifold is a space form. This reasoning is used to justify the Friedmann model in cosmology. The speaker will demonstrate that if a Riemannian manifold is only locally almost isotropic in a way which allows for both weak gravitational lensing and strong gravitational lensing, that the manifold is then Gromov-Hausdorff close to a space with constant sectional curvature and the lengths of the third sides of triangles can be approximated using the formulas from spherical, hyperbolic and Euclidean geometry. The estimates will involve a uniform lower bound on Ricci curvature. A similar result only requiring a packing bound will also be discussed.
